An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces
© The Author(s) 2009
Received: 30 January 2009
Accepted: 15 May 2009
Published: 22 June 2009
The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.
The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer, Glockle and Nonnenmacher , Metzler et al. , Podlubny , Gaul et al. , among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. , Ahmad and Nieto , Ahmad and Otero-Espinar , Belarbi et al. , Belmekki et al , Benchohra et al. [11–13], Chang and Nieto , Daftardar-Gejji and Bhalekar , Figueiredo Camargo et al. , and the monographs of Kilbas et al.  and Podlubny .
Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain , and so forth. the same requirements of boundary conditions. Caputo's fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see [18, 19].
where is the Caputo fractional derivative, , , and are given functions satisfying some assumptions that will be specified later, and is a Banach space with norm .
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics  and cellular systems .
Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra , Benchohra et al. [23, 24], Infante , Peciulyte et al. , and the references therein.
In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel  and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni , Guo et al. , Lakshmikantham and Leela , Mönch , and Szufla .
In this section, we present some definitions and auxiliary results which will be needed in the sequel.
Let be the space of functions , whose first derivative is absolutely continuous.
Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.
Definition 2.1 (see ).
The Kuratowski measure of noncompactness satisfies some properties (for more details see ).
(a) is compact ( is relatively compact).
Here and denote the closure and the convex hull of the bounded set , respectively.
For completeness we recall the definition of Caputo derivative of fractional order.
Definition 2.2 (see ).
for , and as .
Here is the delta function.
Definition 2.3 (see ).
Here and denotes the integer part of .
A map is said to be Carathéodory if
(i) is measurable for each
(ii) is continuous for almost each
For our purpose we will only need the following fixed point theorem and the important Lemma.
holds for every subset of , then has a fixed point.
Lemma 2.6 (see ).
3. Existence of Solutions
Let us start by defining what we mean by a solution of the problem (1.1).
A function is said to be a solution of (1.1) if it satisfies (1.1).
Lemma 3.2 (see ).
if and only if is a solution of the fractional boundary value problem (3.1).
For the forthcoming analysis, we introduce the following assumptions
(H1)The functions satisfy the Carathéodory conditions.
then the boundary value problem (1.1) has at least one solution.
Clearly, the subset is closed, bounded, and convex. We will show that satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.
Let be a sequence such that in . Then, for each ,
maps into itself.
For each , by and (3.8) we have for each
is bounded and equicontinuous.
By Step 2, it is obvious that is bounded.
For the equicontinuity of . Let , and . Then
As , the right-hand side of the above inequality tends to zero.
Now let be a subset of such that .
By (3.8) it follows that , that is, for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 2.5 we conclude that has a fixed point which is a solution of the problem (1.1).
4. An Example
Clearly, conditions (H1),(H2) hold with
which is satisfied for each . Then by Theorem 3.4 the problem (4.1) has a solution on .
The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.
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